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Banach-Tarski Paradox


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I don't really see the point of this video. He is giving a rather detailed construction of the Banach-Tarski paradoxon, but at the same time he doesn't even mention the axiom of choice even once (except on one slide). But without calling out the axiom of choice, showing the Banach-Tarski paradox is pointless.

For illustrating the Banach-Tarski paradox, xkcd does it right:

pumpkin_carving.pnghttp://xkcd.com/804/

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3 hours ago, Tullius said:

I don't really see the point of this video. He is giving a rather detailed construction of the Banach-Tarski paradoxon, but at the same time he doesn't even mention the axiom of choice even once (except on one slide). But without calling out the axiom of choice, showing the Banach-Tarski paradox is pointless.

For illustrating the Banach-Tarski paradox, xkcd does it right:

pumpkin_carving.pnghttp://xkcd.com/804/

Well we usually assume under 'normal' conditions that we have the axiom of choice. This allows us to prove there are unmeasurable sets (and to make the pieces for banach-tarski balls). The point of the video is that your everyday intuition about measure (length, area volume etc) and 'continuum' type sets (real numbers, euclidean space etc) can't be combined without causing conflict (some sets might change their volume if you rotate or translate them for example). Axiom of choice lets you explicitly narrow down some examples (like banach-tarski). without it you can't say such things happen.

It's like saying about any maths video "ahh but they didn't mention the axiom of the empty set" - the point is we make the mathematics as we think it should be, and we have tried to come up with axioms that we can reduce everything to. Not the otherway around! So we could have the axiom of choice because it makes possible banach-tarski (that's not actually why we decide to have it, just an example). Mr banach consipres with mr tarski and come up with this mathematically interesting thing. They find there is no way to reduce its description to combinations of the existing axioms, so they add choice. You can talk about the interesting mathematical thing still without ever talking about choice - it is interesting not just because of it's relation to the axiom of choice!

Edited by jf0
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1 hour ago, jf0 said:

Well we usually assume under 'normal' conditions that we have the axiom of choice. This allows us to prove there are unmeasurable sets (and to make the pieces for banach-tarski balls). The point of the video is that your everyday intuition about measure (length, area volume etc) and 'continuum' type sets (real numbers, euclidean space etc) can't be combined without causing conflict (some sets might change their volume if you rotate or translate them for example). Axiom of choice lets you explicitly narrow down some examples (like banach-tarski). without it you can't say such things happen.

It's like saying about any maths video "ahh but they didn't mention the axiom of the empty set" - the point is we make the mathematics as we think it should be, and we have tried to come up with axioms that we can reduce everything to. Not the otherway around! So we could have the axiom of choice because it makes possible banach-tarski (that's not actually why we decide to have it, just an example). Mr banach consipres with mr tarski and come up with this mathematically interesting thing. They find there is no way to reduce its description to combinations of the existing axioms, so they add choice. You can talk about the interesting mathematical thing still without ever talking about choice - it is interesting not just because of it's relation to the axiom of choice!

I have got to know the Banach-Tarski paradox, like probably many other mathematicians, as a way of the professor showing the first year students that the axiom of choice, unlike the other Zermel-Fraenkel axioms, isn't all that nice. It has some pretty nasty consequences.

At the same time, it is quite intuitive and leads to rather intuitive results. So it has become normal to use it, but, as far as I am concerned, usually it is being announced specifically as being used for such and such theorem or theory. The video not stating it, left me a bit puzzled.

However, that doesn't really make his video bad. Since he had to concentrate the whole discussion into just 20 minutes of video, he chose to present the proof the Banach-Tarski paradoxon (which by the way was quite nice, since while leaving out all of the details, it was very clear). But he had to leave out so much, like the explanation why the result is no problem at all. I guess it falls under the same category as the video of Numberphile, where he extended his notion of convergence to Cesaro convergence, without stating it: It is not wrong, but...

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